Recursion

1
Recursion

• CS 105

2
Iteration revisited

• Iteration or repeated execution of statements are
  typically achieved through loops
• In Java, the the available loop forms are the
  for loop, while loop, and do-while loop
• There is an alternative Recursion(Chapter 13
  of the Horstmann text)

3
Recursion defined

• Recursion occurs when a method calls itself
• Methods can call other methods so it should be
  possible (at least syntactically) for a method to
  invoke itself
• Appears counter-intuitive
• Implies infinite definition
• May result in endless execution
• Analogy mirror in front of another mirror

4
Example 1 factorial

• For loop version
• public static int factorial( int n )
• result 1
• for( int i 1 i lt n i )
• result i
• return result
• This arises from a definition of factorial that
  goes n! 123 n

5
Example 1 factorial

• The factorial function can instead be defined as
  follows
• n! 1 if n 0 n(n-1)! if n gt 1
• 3! 32!, 2! 21!, 1! 10!, 0!1
• Or, fac(3) 3fac(2) 3(2fac(1))
  3(2(1fac(0)) 3(2(1(1)))

6
Example 1 factorial

• Recursive version
• public static int fac( int n )
• if ( n 0 )
• return 1
• else
• return n fac( n-1 )
• Note the importance of the base case (n0) it
  ensures that the recursion collapses and prevents
  endless execution from taking place
• Reminiscent of mathematical induction

7
About recursion

• Advantages of recursion
• More elegant code
• More provable or reliable code since the code
  often closely matches its definition
• Disadvantages
• Possibly slower code (uses and may abuse the
  runtime stack)
• Sometimes difficult to trace through an execution

8
Example 2 power

• Alternative definitions of the power function
• xn xxx x (multiply n times)
• xn 1 if n 0 xxn-1 if n gt 1
• Come up with corresponding loop and recursive
  versions for the above definitions (very similar
  to the factorial example)

9
Example 2 power

• Better (recursive) definition
• xn 1 if n 0 xn/2 xn/2 if n is
  even x x(n-1)/2 x(n-1)/2 if n is odd
• Example
• x13 x x6 x6
• where x6 x3 x3
• where x3 x x1 x1
• where x1 x x0 x0
• where x0 1

10
Example 2 power

• Algorithm Power( x, n )
• Input A number x and integer n gt 0
• Output The value xn
• if n 0 then
• return 1
• if n is odd then
• y ? Power( x, (n-1)/2 )
• return xyy
• if n is even then
• y ? Power( x, n/2 )
• return yy

11
Time complexity analysis

• Count the number of multiplications performed
• Iterative version O(n)
• First recursive version O(n)
• Second recursive version O( log n )

12
Example 3 array reversal

• Algorithm ReverseArray( A,i,j )
• Input An array A, integer indices i and j
• Output The reversal of elements in A from index
  i to j
• if i lt j then
• Swap Ai and Aj
• ReverseArray( A, i1, j-1 )
• Let S be an array (indexed at 0..n-1). A reversal
  of the entire array will occur with the call
  ReverseArray( S, 0, n-1 )
• The base case is invisible occurs when i gt j

13
Summary

• Recursion is an alternative to loops
• Recursion occurs when a method invokes itself
• Often useful when the definition of the problem
  (or its solution) is itself a recurrence
• Make sure that a base case is included so that
  the recursion collapses and the execution
  terminates